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Stability analysis of irreversible chemical reaction-diffusion systems with boundary equilibria

Thi Lien Nguyen, Bao Quoc Tang

Abstract

Large time dynamics of reaction-diffusion systems modeling some irreversible reaction networks are investigated. Depending on initial masses, these networks possibly possess boundary equilibria, where some of the chemical concentrations are completely used up. In the absence of these equilibria, we show an explicit convergence to equilibrium by a modified entropy method, where it is shown that reactions in a measurable set with positive measure is sufficient to combine with diffusion and to drive the system towards equilibrium. When the boundary equilibria are present, we show that they are unstable (in Lyapunov sense) using some bootstrap instability technique from fluid mechanics, while the nonlinear stability of the positive equilibrium is proved by exploiting a spectral gap of the linearized operator and the uniform-in-time boundedness of solutions.

Stability analysis of irreversible chemical reaction-diffusion systems with boundary equilibria

Abstract

Large time dynamics of reaction-diffusion systems modeling some irreversible reaction networks are investigated. Depending on initial masses, these networks possibly possess boundary equilibria, where some of the chemical concentrations are completely used up. In the absence of these equilibria, we show an explicit convergence to equilibrium by a modified entropy method, where it is shown that reactions in a measurable set with positive measure is sufficient to combine with diffusion and to drive the system towards equilibrium. When the boundary equilibria are present, we show that they are unstable (in Lyapunov sense) using some bootstrap instability technique from fluid mechanics, while the nonlinear stability of the positive equilibrium is proved by exploiting a spectral gap of the linearized operator and the uniform-in-time boundedness of solutions.

Paper Structure

This paper contains 8 sections, 16 theorems, 166 equations, 1 table.

Table of Contents

  1. Introduction
  2. Catalytic-type reactions
  3. Stability of positive equilibrium - Entropy method
  4. Stability of boundary equilibrium - Linearization
  5. Instability of boundary equilibrium - Bootstrap instability
  6. Symmetric irreversible reactions
  7. Stability of positive equilibrium
  8. Instability of boundary equilibrium

Key Result

Theorem 2.1

Assume A1--A2. Then there exists a unique global classical solution in the following sense: for any $T>0$, satisfying each equation in p1 pointwise. Moreover, this solution is bounded uniformly in time, i.e. there is a positive constant $\mathcal{K}$ such that

Theorems & Definitions (32)

  • Theorem 2.1
  • proof
  • Lemma 2.2
  • Remark 2.3
  • proof
  • Theorem 2.4
  • proof
  • Proposition 2.5
  • proof
  • Theorem 2.6
  • ...and 22 more